Topological energy bands. The topological identity of energy bands can be characterized by non-zero Chern numbers of bulk bands or edge states in a finite-size system. Within our model, considering a strip sample, we draw edge states as shown in Fig. REF a. In the figure, the dense overlapped states are bulk states, which are also the projection of energy bands in the \(k_y\)  direction. The states lying in the gaps and connecting two bulk bands are edge states.
In the absence of DM interaction, the first (second) band touches the third (fourth) band at the \(\text{M}(\pi ,\pi )\)  (\(\Gamma (0,0)\) ) point. The intermediate energy bands are flat at those high symmetry points. With a nonzero DM term, we can open gaps at those degenerate points to turn the system into a magnon insulator [1]}. The gap expressions are \(\pm 2(D_1+2D_2)S\)  or \( \pm 2|D_1-2D_2|S\)  at the \(\Gamma \)  and the \(\text{M}\)  points, respectively. See Supplementary Information for more detail. Note, Chern numbers are well-defined in band insulators, only.
